Using effective homological algebra for factoring and decomposing of - - PowerPoint PPT Presentation

Using effective homological algebra for factoring and decomposing of linear functional systems

Thomas Cluzeau & Alban Quadrat {Thomas.Cluzeau, Alban.Quadrat}@inria.fr

INRIA Sophia Antipolis, CAFE Project, 2004 route des lucioles, BP 93, 06902 Sophia Antipolis cedex, France. www-sop.inria.fr/cafe/Alban.Quadrat/index.html

S´ eminaire ALGO, INRIA Rocquencourt, 26/06/2006

Alban Quadrat Factoring and decomposing linear functional systems

Introduction: factorization and decomposition

• Let L(∂) be a scalar ordinary or partial differential operator.
• When is it possible to find L1(∂) and L2(∂) such that:

L(∂) = L2(∂) L1(∂)?

• We note that L1(∂) y = 0 ⇒ L(∂) y = 0.
• L(∂) y = 0 is equivalent to the cascade integration:

L1(∂) y = z & L2(∂) z = 0.

• When is the integration of L(∂) y = 0 equivalent to:

L2(∂) z = 0 & L1(∂) u = 0? (L1 X + Y L2 = 1 ⇒ L1(X z) = z ⇒ y = u + X z)

Alban Quadrat Factoring and decomposing linear functional systems

Introduction: factorization and decomposition

• Let us consider the first order ordinary differential system:

∂ y = E(t) y, E(t) ∈ k(t)p×p. (⋆)

• When does it exist an invertible change of variables

y = P(t) z, such that (⋆) ⇔ ∂ z = F(t) z, where F = −P−1 (∂P − E P) is either of the form: F = F11 F12 F22

• r

F = F11 F22

• ?

Alban Quadrat Factoring and decomposing linear functional systems

Factorization: known cases

Square differential systems: Beke’s algorithm (Beke1894, Schwarz89, Bronstein94, Tsar¨

• ev94. . . )

Eigenring (Singer96, Giesbrecht98, Barkatou-Pfl¨ ugel98, Barkatou01 - ideas in Jacobson37. . . ) Square (q-)difference systems (generalizations): Barkatou01, Bomboy01. . . Square D-finite partial differential systems (connections): Li-Schwarz-Tsar¨ ev03, Wu05. . . Same cases in positive characteristic and modular approaches: van der Put95, C.03, Giesbrecht-Zhang03, C.-van Hoeij04,06, Barkatou-C.-Weil05. . .

Alban Quadrat Factoring and decomposing linear functional systems

General Setting

What about general linear functional systems?

• Example (Saint Venant equations): linearized model around the

Riemann invariants (Dubois-Petit-Rouchon, ECC99):

• y1(t − 2 h) + y2(t) − 2 ˙

y3(t − h) = 0, y1(t) + y2(t − 2 h) − 2 ˙ y3(t − h) = 0.

• Let D = R

d

dt , δ

• and consider the system matrix:

R =

• δ2

1 −2 δ d

dt

1 δ2 −2 δ d

dt

• ∈ D2×3.

Question: ∃ U ∈ GL3(D), V ∈ GL2(D) such that: V R U =

• α1

α2 α3

• , α1, α2, α3 ∈ D?

Alban Quadrat Factoring and decomposing linear functional systems

Outline

• Type of systems: Partial differential/discrete/differential

time-delay. . . linear systems (LFSs).

• General topic: Algebraic study of linear functional systems

(LFSs) coming from mathematical physics, engineering sciences. . .

• Techniques: Module theory and homological algebra.
• Applications: Equivalences of systems, Galois symmetries,

quadratic first integrals/conservation laws, decoupling problem. . .

• Implementation: package morphisms based on OreModules:

http://wwwb.math.rwth-aachen.de/OreModules.

Alban Quadrat Factoring and decomposing linear functional systems

General methodology

1 A linear system is defined by means of a matrix R with entries

in a ring D of functional operators: R y = 0. (⋆)

2 We associate a finitely presented left D-module M with (⋆). 3 A dictionary exists between the properties of (⋆) and M. 4 Homological algebra allows us to check properties of M. 5 Effective algebra (non-commutative Gr¨

• bner/Janet bases)

leads to constructive algorithms.

6 Implementation (Maple, Singular/Plural, Cocoa. . . ). Alban Quadrat Factoring and decomposing linear functional systems

• I. Ore Module associated with a linear functional system

Alban Quadrat Factoring and decomposing linear functional systems

Ore algebras

Consider a ring A, an automorphism σ of A and a σ-derivation δ: δ(a b) = σ(a) δ(b) + δ(a) b. Definition: A non-commutative polynomial ring D = A[∂; σ, δ] in ∂ is called skew if ∀ a ∈ A, ∂ a = σ(a) ∂ + δ(a). Definition: Let us consider A = k, k[x1, . . . , xn] or k(x1, . . . , xn). The skew polynomial ring D = A[∂1; σ1, δ1] . . . [∂m; σm, δm] is called an Ore algebra if we have:

• σi δj = δj σi,

1 ≤ i, j ≤ m, σi(∂j) = ∂j, δi(∂j) = 0, j < i. ⇒ D is generally a non-commutative polynomial ring.

Alban Quadrat Factoring and decomposing linear functional systems

Examples of Ore algebras

• Partial differential operators: A = k, k[x1, . . . , xn], k(x1, . . . , xn),

D = A

• ∂1; id,

∂ ∂x1

• . . .
• ∂n; id,

∂ ∂xn

• ,

P =

0≤|µ|≤m aµ(x) ∂µ ∈ D,

∂µ = ∂µ1

1 . . . ∂µn n .

• Shift operators:

D = A[∂; σ, 0], A = k, k[n], k(n), P = m

i=0 ai(n) ∂i ∈ D,

σ(a)(n) = a(n + 1).

• Differential time-delay operators:

D = A

• ∂1; id, d

dt

• [∂2; σ, 0],

A = k, k[t], k(t), P =

0≤i+j≤m aij(t) ∂i 1 ∂j 2 ∈ D.

Alban Quadrat Factoring and decomposing linear functional systems

Exact sequences

• Definition: A sequence of D-morphisms M′

f

− → M

g

− → M′′ is said to be exact at M if we have: ker g = im f .

• Example: If f : M −

→ M′ is a D-morphism, we then have the following exact sequences:

1 0 −

→ ker f

i

− → M

ρ

− → coim f M/ ker f − → 0.

2 0 −

→ im f

j

− → M′

κ

− → coker f M′/im f − → 0.

3 0 −

→ ker f

i

− → M

f

− → M′

κ

− → coker f − → 0.

Alban Quadrat Factoring and decomposing linear functional systems

A left D-module M associated with R η = 0

• Let D be an Ore algebra, R ∈ Dq×p and a left D-module F.
• Let us consider kerF(R.) = {η ∈ Fp | R η = 0}.
• As in number theory or algebraic geometry, we associate with

the system kerF(R.) the finitely presented left D-module: M = D1×p/(D1×q R).

• Malgrange’s remark: applying the functor homD(., F) to the

finite free resolution (exact sequence) D1×q

.R

− → D1×p

π

− → M − → 0, λ = (λ1, . . . , λq) − → λ R we then obtain the exact sequence: Fq

R.

← − Fp

π⋆

← − homD(M, F) ← − 0. R η ← − η = (η1, . . . , ηp)T

Alban Quadrat Factoring and decomposing linear functional systems

Example: Linearized Euler equations

• The linearized Euler equations for an incompressible fluid can be

defined by the system matrix R =     ∂1 ∂2 ∂3 ∂t ∂1 ∂t ∂2 ∂t ∂3     ∈ D4×4, where D = R

• ∂1, id,

∂ ∂x1

∂2, id,

∂ ∂x2

∂3, id,

∂ ∂x3

∂t, id, ∂

∂t

• .
• Let us consider the left D-module F = C∞(Ω) (Ω open convex

subset of R4) and the D-module: M = D1×4/(D1×4 R). The solutions of R y = 0 in F are in 1 − 1 correspondence with the morphisms from M to F, i.e., with the elements of: homD(M, F).

Alban Quadrat Factoring and decomposing linear functional systems

• II. Morphisms between Ore modules finitely presented

by two matrices R and R′ of functional operators

Alban Quadrat Factoring and decomposing linear functional systems

Morphims of finitely presented modules

• Let D be an Ore algebra of functional operators.
• Let R ∈ Dq×p, R′ ∈ Dq′×p′ be two matrices.
• Let us consider the finitely presented left D-modules:

M = D1×p/(D1×q R), M′ = D1×p′/(D1×q′ R′).

• We are interested in the abelian group homD(M, M′) of

D-morphisms from M to M′: D1×q

.R

− → D1×p

π

− → M − → 0 ↓ f D1×q′

.R′

− − → D1×p′

π′

− → M′ − → 0.

Alban Quadrat Factoring and decomposing linear functional systems

Morphims of finitely presented modules

• Let D be an Ore algebra of functional operators.
• Let R ∈ Dq×p, R′ ∈ Dq′×p′ be two matrices.
• We have the following commutative exact diagram:

D1×q

.R

− → D1×p

π

− → M − → 0 ↓ .Q ↓ .P ↓ f D1×q′

.R′

− − → D1×p′

π′

− → M′ − → 0. ∃ f : M → M′ ⇐ ⇒ ∃ P ∈ Dp×p′, Q ∈ Dq×q′ such that: R P = Q R′. Moreover, we have f (π(λ)) = π′(λ P), for all λ ∈ D1×p.

Alban Quadrat Factoring and decomposing linear functional systems

Eigenring: ∂y = E y & ∂z = F z

• D = A[∂; σ, δ],

E, F ∈ Ap×p, R = ∂ Ip − E, R′ = ∂ Ip − F. 0 − → D1×p

.(∂p I−E)

− − − − − − → D1×p

π

− → M − → 0 ↓ .Q ↓ .P ↓ f 0 − → D1×p

.(∂ Ip−F)

− − − − − → D1×p

π′

− → M′ − → 0. (∂ Ip − E) P = Q (∂ Ip − F) ⇐ ⇒

• σ(P) = Q ∈ Ap×p,

δ(P) = E P − σ(P) F. If P ∈ Ap×p is invertible, we then have: F = −σ(P)−1(δ(P) − E P).

Alban Quadrat Factoring and decomposing linear functional systems

Eigenring: ∂y = E y & ∂z = F z

• D = A[∂; σ, δ],

E, F ∈ Ap×p, R = ∂ Ip − E, R′ = ∂ Ip − F. 0 − → D1×p

.(∂p I−E)

− − − − − − → D1×p

π

− → M − → 0 ↓ .Q ↓ .P ↓ f 0 − → D1×p

.(∂ Ip−F)

− − − − − → D1×p

π′

− → M′ − → 0. (∂ Ip − E) P = Q (∂ Ip − F) ⇐ ⇒

• σ(P) = Q ∈ Ap×p,

δ(P) = E P − σ(P) F. If P ∈ Ap×p is invertible, we then have: F = −σ(P)−1(δ(P) − E P).

• Differential case: δ = d

dt , σ = id:

˙ P = E P − P F, F = −P−1( ˙ P − E P).

Alban Quadrat Factoring and decomposing linear functional systems

Eigenring: ∂y = E y & ∂z = F z

• D = A[∂; σ, δ],

E, F ∈ Ap×p, R = ∂ Ip − E, R′ = ∂ Ip − F. 0 − → D1×p

.(∂p I−E)

− − − − − − → D1×p

π

− → M − → 0 ↓ .Q ↓ .P ↓ f 0 − → D1×p

.(∂ Ip−F)

− − − − − → D1×p

π′

− → M′ − → 0. (∂ Ip − E) P = Q (∂ Ip − F) ⇐ ⇒

• σ(P) = Q ∈ Ap×p,

δ(P) = E P − σ(P) F. If P ∈ Ap×p is invertible, we then have: F = −σ(P)−1(δ(P) − E P).

• Discrete case: δ = 0, σ(k) = k − 1:
• E(k) P(k) − P(k − 1) F(k) = 0,

B = σ(P)−1 E P.

Alban Quadrat Factoring and decomposing linear functional systems

Computation of homD(M, M′)

• Problem: Given R ∈ Dq×p and R′ ∈ Dq′×p′, find P ∈ Dp×p′ and

Q ∈ Dq×q′ satisfying the commutation relation R P = Q R′.

• If D is a commutative ring, then homD(M, M′) is a D-module.
• The Kronecker product of E ∈ Dq×p and F ∈ Dr×s is:

E ⊗ F =    E11 F . . . E1p F . . . . . . . . . Eq1 F . . . Eqp F    ∈ D(q r)×(p s). Lemma: If U ∈ Da×b, V ∈ Db×c and W ∈ Dc×d, then we have: U V W = (V1 . . . Vb) (UT ⊗ W ). R P Ip′ = (P1 . . . Pp) (RT ⊗ Ip′), Iq Q R′ = (Q1 . . . Qq) (Iq ⊗ R′). We are reduced to compute kerD

• .

RT ⊗ Ip′ −Iq ⊗ R′

• .

Alban Quadrat Factoring and decomposing linear functional systems

Computation of homD(M, M′)

• Problem: Given R ∈ Dq×p and R′ ∈ Dq′×p′, find P ∈ Dp×p′ and

Q ∈ Dq×q′ satisfying the commutation relation R P = Q R′.

• If D is a non-commutative ring, then homD(M, M′) is an abelian

group and generally an infinite-dimensional k-vector space. ⇒ find a k-basis of morphisms with given degrees in xi and in ∂j:

1 Take an ansatz for P with chosen degrees. 2 Compute R P and a Gr¨

• bner basis G of the rows of R′.

3 Reduce the rows of R P w.r.t. G. 4 Solve the system on the coefficients of the ansatz so that all

the normal forms vanish.

5 Substitute the solutions in P and compute Q by means of a

factorization.

Alban Quadrat Factoring and decomposing linear functional systems

Example: Bipendulum

• We consider the Ore algebra D = R(g, l)

d

dt

• .
• We consider the matrix of the bipendulum with l = l1 = l2:

R =

• d2

dt2 + g l

−g

l d2 dt2 + g l

−g

l

• ∈ D2×3.
• Let us consider the D-module M = D1×3/(D1×2 R).
• We obtain that endD(M) is defined by the matrices:

P =   α1 α2 α3 g α4 α1 + α2 − α4 α3 g α3 D2 l + α1 + α2 + α3 g   , Q = α1 α2 α4 α1 + α2 − α4

• ,

∀ α1, . . . , α4 ∈ D.

Alban Quadrat Factoring and decomposing linear functional systems

Example: Bomboy’s PhD, p. 80

q-dilatation case: D = R(q)(x) [H] where H(f (x)) = f (q x) and: R =   H −1 −1 − q3 x2 1 − q x2 x (1 − q2) 1 − q x2 + H   ∈ D2×2.

• Searching for endomorphisms with degree 0 in H and 2 in x

(both in numerator and denominator), we obtain

P =      −a + b x q − b x + a q x2 c (−1 + q x2) b (−1 + x2) c (−1 + q x2) b (−1 + q2 x2) c (−1 + q x2) −a + b x q − b x − a q x2 c (−1 + q x2)      ,

where, a, b, c are constants or P = I2 (and corresponding Q).

Alban Quadrat Factoring and decomposing linear functional systems

Saint-Venant equations

• Let D = Q
• ∂1; id, d

dt

• [∂2; σ, 0] be the ring of differential

time-delay operators and consider the matrix of the tank model:

R =

• ∂2

2

1 −2 ∂1 ∂2 1 ∂2

2

−2 ∂1 ∂2

• ∈ D2×3.
• The endomorphisms of M = D1×3/(D1×2 R) are defined by:

Pα =    α1 α2 + 2 α4 ∂1 + 2 α5 ∂1 ∂2 α4 ∂2 + α5 α2 2 α3 ∂1 ∂2 α1 − 2 α4 ∂1 − 2 α5 ∂1 ∂2 2 α3 ∂1 ∂2 −α4 ∂2 − α5 α1 + α2 + α3 (∂2

2 + 1)

   , Qα =

• α1 − 2 α4 ∂1

α2 + 2 α4 ∂1 α2 + 2 α5 ∂1 ∂2 α1 − 2 α5 ∂1 ∂2

• ,

∀ α1, . . . , α5 ∈ D.

Alban Quadrat Factoring and decomposing linear functional systems

Euler-Tricomi equation

• Let us consider the Euler-Tricomi equation (transonic flow):

∂2

1 u(x1, x2) − x1 ∂2 2 u(x1, x2) = 0.

• Let D = A2(Q), R = (∂2

1 − x1 ∂2 2) ∈ D and M = D/(D R).

endD(M)1,1 is defined by:

• P = a1 + a2 ∂2 + 3

2 a3 x2 ∂2 + a3 x1 ∂1,

Q = (a1 + 2 a3) + a2 ∂2 + 3

2 a3 x2 ∂2 + a3 x1 ∂1,

endD(M)2,0 is defined by P = Q = a1 + a2 ∂2 + a3 ∂2

2.

endD(M)2,1 is defined by:          P = a1 + a2 ∂2 + 3

2 a3 x2 ∂2 + a3 x1 ∂1

+a4 ∂2

2 + 3 2 a5 x2 ∂2 2 + a5 x1 ∂1 ∂2,

Q = (a1 + 2 a3) + a2 ∂2 + 3

2 a3 x2 ∂2 + a3 x1 ∂1

+a4 ∂2

2 + a5 x1 ∂1 ∂2 + 2 a5 ∂2 + 3 2 a5 x2 ∂2 2.

Alban Quadrat Factoring and decomposing linear functional systems

• III. A few applications:

Galois symmetries, quadratic first integrals of motion and conservation laws

Alban Quadrat Factoring and decomposing linear functional systems

Galois Symmetries

We have the following commutative exact diagram: D1×q

.R

− → D1×p

π

− → M − → 0 ↓ .Q ↓ .P ↓ f D1×q′

.R′

− − → D1×p′

π′

− → M′ − → 0. (⋆) If F is a left D-module, by applying the functor homD(·, F) to (⋆), we then obtain the following commutative exact diagram: 0 = Q (R′ y) = R (P y) ← − P y Fq

R.

← − Fp ← − kerF(R.) ← − 0 ↑ Q. ↑ P. ↑ f ⋆ Fq′

R′.

← − − Fp′ ← − kerF(R′.) ← − 0. 0 = R′ y ← − y ⇒ f ⋆ sends kerF(R.′) to kerF(R.) (R′ = R: Galois symmetries).

Alban Quadrat Factoring and decomposing linear functional systems

Example: Linear elasticity

• Consider the Killing operator for the euclidian metric defined by:

R =    ∂1 ∂2/2 ∂1/2 ∂2    .

• The system R y = 0 admits the following general solution:

y =

• c1 x2 + c2

−c1 x1 + c3

• ,

c1, c2, c3 ∈ R. (⋆)

• We find that endD(D1×2/(D1×3 R)) is defined by:

P =

• α1

α2 ∂1 2 α3 ∂1 + α1

• ,

α1, α2, α3 ∈ D.

• Applying P to (⋆), we then get the new solution:

y = P y =

• α1 c1 x2 + α1 c2 − α2 c1

−α1 c1 x1 + α1 c3 − 2 α3 c1

• , i.e., R y = 0.

Alban Quadrat Factoring and decomposing linear functional systems

Quadratic first integrals of motion

Let us consider a morphism f from N to M defined by: 0 − → D1×p

.(∂ Ip+E T )

− − − − − − → D1×p

π

− →

• N

− → 0 ↓ .P ↓ .P ↓ f 0 − → D1×p

.(∂ Ip−E)

− − − − − → D1×p

π′

− → M − → 0. We then have: ˙ P + E T P + P E = 0. If V (x) = xT P x, then ˙ V (x) = xT ( ˙ P + E T P + P E) x so that: ˙ P + E T P + P E = 0 ⇐ ⇒ V (x) = xT P x first integral. ⇒ Morphisms from N to M give quadratic first integrals. If E is a skew-symmetric matrix, i.e., E = −E T, then we have: (∂ Ip + E T) = (∂ Ip − E),

• N = M,

homD( N, M) = endD(M).

Alban Quadrat Factoring and decomposing linear functional systems

Example: Landau & Lifchitz (p. 117)

• Consider R = ∂ I4 − E, where E =

    1 −ω2 α 1 −ω2 α    .

• We find that the morphisms from

N to M are defined by

P =     c1 ω4 c2 ω2 −ω2 (c1 α + c2) c1 ω2 −c2 ω2 c1 ω2 −c1 ω2 + c2 α −c2 −ω2 (c1 α − c2) −c1 ω2 − c2 α c1 (α2 + ω2) −c1 α + c2 c1 ω2 c2 −c1 α − c2 c1     ,

which leads to the quadratic first integral V (x) = xT P x: V (x) = c1 ω4 x1(t)2 − 2 x1(t) ω2 x3(t) c1 α + 2 x1(t) c1 ω2 x4(t) +x2(t)2 c1 ω2 − 2 x2(t) c1 x3(t) ω2 + c1 x3(t)2 α2 +c1 x3(t)2 ω2 − 2 x3(t) x4(t) c1 α + c1 x4(t)2.

Alban Quadrat Factoring and decomposing linear functional systems

Formal adjoint

• Let D = A
• ∂1; id,

∂ ∂x1

• . . .
• ∂n; id,

∂ ∂xn

• be the ring of differential
• perators with coefficients in A (e.g., k[x1, . . . , xn], k(x1, . . . , xn)).
• The formal adjoint

R ∈ Dp×q of R ∈ Dq×p is defined by: < λ, R η >=< R λ, η > +

n

• i=1

∂i Φi(λ, η).

• The formal adjoint

R can be defined by R = (θ(Rij))T ∈ Dp×q, where θ : D → D is the involution defined by:

1 ∀ a ∈ A,

θ(a) = a.

2 θ(∂i) = −∂i,

i = 1, . . . , n. Involution: θ2 = idD, ∀ P1, P2 ∈ D: θ(P1 P2) = θ(P2) θ(P1).

Alban Quadrat Factoring and decomposing linear functional systems

Conservation laws

• Let us consider the left D-modules:

M = D1×p/(D1×q R),

• N = D1×q/(D1×p

R).

• Let f :

N → M be a morphism defined by the matrices P and Q.

• Let F be a left D-module and the commutative exact diagram:

Fp

e R.

← − Fq ← − kerF( R.) ← − 0 ↑ Q. ↑ P. ↑ f ⋆ Fq

R.

← − Fp ← − kerF(R.) ← − 0.

• η ∈ Fp solution of R η = 0 ⇒ λ = P η is a solution of

R λ = 0. ⇒ < P η, R η > − < R (P η), η >=

n

• i=1

∂i Φi(P η, η) = 0, i.e., Φ = (Φ1(P η, η), . . . , Φn(P η, η))T satisfies div Φ = 0.

Alban Quadrat Factoring and decomposing linear functional systems

Example: Laplacian operator

• Let us consider the Laplacian operator ∆ y(x1, x2) = 0, where:

∆ = ∂2

1 + ∂2 2 ∈ D = Q

• ∂1; id, ∂

∂x1 ∂2; id, ∂ ∂x2

• .
• The formal adjoint

R of R is then defined by: λ (∆ η) − (∆ λ) η = ∂1 (λ (∂1 η) − (∂1 λ) η) + ∂2 (λ (∂2 η) − (∂2 λ) η).

• R = ∆ =

R ∈ D ⇒ homD( N, M) = endD(M) = D.

• if F is a D-module (e.g., C ∞(Ω)), then we have:

∀ α ∈ D, ∀ η ∈ kerF(∆.), λ = α y ∈ kerF(∆.). ⇒ div Φ = ∂1 Φ1+∂2 Φ2 = 0, Φ =

• (α y) (∂1 y) − y (∂1 α y)

(α y) (∂2 y) − y (∂2 α y)

• .

Alban Quadrat Factoring and decomposing linear functional systems

• IV. Factorization of linear functional systems

Alban Quadrat Factoring and decomposing linear functional systems

Kernel and factorization

λ − → y D1×q

.R

− → D1×p

π

− → M − → 0 ↓ .Q ↓ .P ↓ f D1×q′

.R′

− − → D1×p′

π′

− → M′ − → 0 ∃ µ − → µ R = λ P − →

• kerD
• .

P R′

• = D1×r (S

− T) ⇒ {λ ∈ D1×p | λ P ∈ D1×q R} = D1×r S ⇒ ker f = (D1×r S)/(D1×q R).

• (D1×q (R

− Q)) ∈ kerD

• .

P R′

• ⇒ (D1×q R) ⊆ (D1×r S).

∃ V ∈ Dq×r : R = V S.

Alban Quadrat Factoring and decomposing linear functional systems

Kernel and factorization

We have the following commutative exact diagram: ↓ ker f ↓ i D1×q

.R

− → D1×p

π

− → M − → 0 ↓ .V

• ↓ κ

D1×r

.S

− → D1×p

π′

− → M/ ker f − → 0. ↓

Alban Quadrat Factoring and decomposing linear functional systems

Example: Linearized Euler equations

• Let R =

    ∂1 ∂2 ∂3 ∂t ∂1 ∂t ∂2 ∂t ∂3     over D = R[∂1, ∂2, ∂3, ∂t].

• Let us consider f ∈ endD(M) defined by:

P =     ∂2

3

−∂2 ∂3 −∂2 ∂3 ∂2

2

    .

• Computing kerD
• .

P R

• and factorizing R by S, we obtain:

S =       1 ∂2 ∂3 −∂t ∂t 1       , V =     ∂1 1 ∂t ∂1 −1 ∂2 1 ∂3     .

Alban Quadrat Factoring and decomposing linear functional systems

Example: Linearized Euler equations

• We have R = V S where:

    ∂1 ∂2 ∂3 ∂t ∂1 ∂t ∂2 ∂t ∂3     =     ∂1 1 ∂t ∂1 −1 ∂2 1 ∂3           1 ∂2 ∂3 −∂t ∂t 1       .

• The solutions of S y = 0 are particular solutions of R y = 0.

⇒ Integrating S, we obtain the following solutions of R y = 0:

y(x1, x2, x3, t) =       − ∂

∂x3 ξ(x1, x2, x3) ∂ ∂x2 ξ(x1, x2, x3)

      , ∀ ξ ∈ C ∞(Ω).

Alban Quadrat Factoring and decomposing linear functional systems

Free modules & similarity transformations

• Definition: A left D-module M is free if there exists l ∈ Z+ s.t.:

M ∼ = D1×l.

• Proposition: Let P ∈ Dp×p. We have the equivalences:

1 kerD(.P) and coimD(.P) are free left D-modules of rank p

and p − m.

2 There exists a unimodular matrix U ∈ Dp×p, i.e.,

U ∈ GLp(D), such that: J U P U−1 = J2

• ,

J2 ∈ D(p−m)×p. ⇒ U = (UT

1

UT

2 )T,

• kerD(.P) = D1×m U1

coimD(.P) = π′(D1×(p−m) U2).

Alban Quadrat Factoring and decomposing linear functional systems

A useful proposition

• Proposition: Let R ∈ Dq×p and P ∈ Dp×p, Q ∈ Dq×q be two

matrices satisfying: R P = Q R. Let U ∈ GLp(D) and V ∈ GLq(D) such that

• P = U−1 JP U,

Q = V −1 JQ V , for certain JP ∈ Dp×p and JQ ∈ Dq×q. Then, the matrix R = V R U−1 satisfies: R JP = JQ R.

Alban Quadrat Factoring and decomposing linear functional systems

A commutative diagram

The following commutative diagram D1×q D1×p D1×q D1×p D1×q D1×p D1×q D1×p

• .JQ
• .V
• .(V R U−1)
• .Jp
• .U
• .Q
• .R
• .P
• .V
• .(V R U−1)
• .U
• .R

implies (V R U−1) Jp = JQ (V R U−1).

Alban Quadrat Factoring and decomposing linear functional systems

Block triangular decomposition

• Theorem: Let R ∈ Dq×p, M = D1×p/(D1×q R) and

f ∈ endD(M) defined by P and Q satisfyin R P = Q R. If the left D-modules kerD(.P), coimD(.P), kerD(.Q), coimD(.Q) are free of rank m, p − m, l, q − l, then there exist two matrices U = (UT

1

UT

2 )T ∈ GLp(D) and V = (V T 1

V T

2 )T ∈ GLq(D)

such that R = V R U−1 = V1 R W1 V2 R W1 V2 R W2

• ∈ Dq×p,

where U−1 = (W1 W2), W1 ∈ Dp×m, W2 ∈ Dp×(p−m) and: U1 ∈ Dm×p, U2 ∈ D(p−m)×p, V1 ∈ Dl×q, V2 ∈ D(q−l)×q.

Alban Quadrat Factoring and decomposing linear functional systems

Example: OD system

• Let D = k[t]
• ∂; id, d

dt

• and M = D1×4/(D1×4 R), where:

R =     ∂ −t t ∂ ∂ t ∂ − t ∂ −1 ∂ −t ∂ + t ∂ − 1 ∂ ∂ − t t ∂     ∈ D4×4.

• An endomorphism f of M is defined by:

P =     1 1     , Q =     t + 1 1 −1 −t 1 1 −1 t + 1 1 −1 −t t 1 −1 −t + 1     .

• We can prove that the left D-modules kerD(.P), coimD(.P),

kerD(.Q) and coimD(.Q) are free of rank 2.

Alban Quadrat Factoring and decomposing linear functional systems

Example: OD system

• We obtain:

       U1 = 1 1

• ,

U2 = 1 1

• ,

V1 = 1 −1 1 t − 1 −t

• ,

V2 = 1 −1 1

• .

⇒ U =     1 1 1 1     , V =     1 −1 1 t − 1 −t 1 −1 1     , we then obtain that R is equivalent to: R = V R U−1 =      −∂ 1 t ∂ − t −∂ − t ∂ + t ∂ − 1 ∂ −t −∂ 1 ∂      .

Alban Quadrat Factoring and decomposing linear functional systems

Example: Saint-Venant equations

• We consider D = Q
• ∂1; id, d

dt

• [∂2; σ, 0] and:

R =

• ∂2

2

1 −2 ∂1 ∂2 1 ∂2

2

−2 ∂1 ∂2

• .
• A endomorphism f of M is defined by:

P =    2 ∂1 ∂2 −2 ∂1 ∂2 1 −1    , Q =

• 2 ∂1 ∂2

−2 ∂1 ∂2

• .
• We can check that kerD(.P), coimD(.P), kerD(.Q) and

coimD(.Q)) are free D-modules of rank respectively 2, 1, 1, 1. ⇒      U1 =

• 1

−1 2 ∂1 ∂2

• ,

U2 = (0 1), V1 = (1 0), V2 = (0 1).

Alban Quadrat Factoring and decomposing linear functional systems

Example: Saint-Venant equations

• If we denote by

U =    1 −1 2 ∂1 ∂2 1    , V =

• 1

1

• ,

we then obtain U−1 =    1 −1 2 ∂1 ∂2 1    ∈ GL3(D), and the matrix R is equivalent to: R = V R U−1 =

• ∂2

1

−1 1 −∂2

1

2 ∂1 ∂2 (∂2

1 − 1)

• .

Alban Quadrat Factoring and decomposing linear functional systems

Ker f , im f , coim f and coker f

• Proposition: Let M = D1×p/(D1×q R), M′ = D1×p′/(D1×q′ R′)

and f : M − → M′ be a morphism defined by R P = Q R′. Let us consider the matrices S ∈ Dr×p, T ∈ Dr×q′, U ∈ Ds×r and V ∈ Dq×r satisfying R = V S, kerD(.S) = D1×s U and: kerD

• .

P R′

• = D1×r (S

− T). Then, we have: ker f = (D1×r S)/(D1×q R) ∼ = D1×l/

• D1×(q+s)

U V

• ,

coim f M/ ker f = D1×p/(D1×r S), im f = D1×(p+q′) P R′

• /(D1×q R) ∼

= D1×p/(D1×r S), coker f M′/im f = D1×p/

• D1×(p+q′)

P R′

• .

Alban Quadrat Factoring and decomposing linear functional systems

Equivalence of systems

• Corollary: Let us consider f ∈ homD(M, M′). Then, we have:

1 f is injective iff one of the assertions holds:

There exists L ∈ Dr×q such that S = L R.

• U

V

• admits a left-inverse.

2 f is surjective iff

P R′

• admits a left-inverse.

3 f is an isomorphism, i.e., M ∼

= M′, iff 1 and 2 are satisfied.

Alban Quadrat Factoring and decomposing linear functional systems

Pommaret’s example

• Equivalence of the systems defined by the following R and R′?

R =

• ∂2

1 ∂2 2 − 1

−∂1 ∂3

2 − ∂2 2

∂3

1 ∂2 − ∂2 1

−∂2

1 ∂2 2

• ,

R′ = (∂1 ∂2 − 1 − ∂2

2).

• We find a morphism given by P =

1 1

• , Q =

1 + ∂1 ∂2 ∂2

1

• .
• U

V

• =
• 1 + ∂1 ∂2

∂2

1

• admits the left-inverse (1 − ∂1 ∂2

∂2

2).

• P

R′

• =

  1 1 ∂1 ∂2 − 1 −∂2

2

  admits the left-inverse (I2

0). ⇒ M = D1×2/(D1×2 R) ∼ = M′ = D1×2/(D R′).

Alban Quadrat Factoring and decomposing linear functional systems

• V. Decomposition of linear functional systems

Alban Quadrat Factoring and decomposing linear functional systems

Projectors of endD(M)

• Lemma: An endomorphism f of M = D1×p/(D1×q R), defined

by the matrices P and Q, is a projector, i.e., f 2 = f , iff there exist Z ∈ Dp×q and Z ′ ∈ Dq×t such that

• P2 = P + Z R,

Q2 = Q + R Z + Z ′ R2, where R2 ∈ Dt×q satisfies kerD(.R) = D1×t R2.

• Some projectors of endD(M) can be computed when a family of

endomorphisms of M is known.

• Example: D = A1(Q), R = (∂2

− t ∂ − 1), M = D1×2/(D R). P =

• −(t + a) ∂ + 1

t2 + a t 1

• ,

P2 = P + (t + a)2

• R.

Alban Quadrat Factoring and decomposing linear functional systems

Projectors of endD(M) & Idempotents

• Particular case: (R2 = 0 and P2 = P) =

⇒ Q2 = Q.

• Lemma: Let us suppose that R2 = 0 and P2 = P + Z R. If there

exists a solution Λ ∈ Dp×q of the Riccatti equation Λ R Λ + (P − Ip) Λ + Λ Q + Z = 0, (⋆) then the matrices P = P + Λ R and Q = Q + R Λ satisfy: R P = Q R, P

2 = P,

Q

2 = Q.

• Example: Λ = (a t

a ∂ − 1)T is a solution of (⋆) ⇒ P = a t ∂2 − (t + a) ∂ + 1 t2 (1 − a ∂) (a ∂ − 1) ∂2 −a t ∂2 + (t − 2 a) ∂ + 2

• , Q = 0,

then satisfy P

2 = P and Q 2 = Q.

Alban Quadrat Factoring and decomposing linear functional systems

Projectors of endD(M)

• Proposition: f is a projector of endD(M), i.e., f 2 = f , iff there

exists a matrix X ∈ Dp×s such that P = Ip − X S and we have the following commutative exact diagram: ↓ ker f ↓ i D1×q

.R

− → D1×p

π

− → M − → 0

.T ↑↓ .V .P ↑↓ .Ip

f ↑↓ κ D1×s

.U

− → D1×r

.S

− → D1×p

π′

− → M/ ker f − → 0.

.X

← − ↓ ⇒ M ∼ = ker f ⊕ im f & S − S X S = T R. (⋆)

• Corollary: If kerD(.S) = 0, then R = V S satisfies:

S X − T V = Ir.

Alban Quadrat Factoring and decomposing linear functional systems

Decomposition of solutions

• Corollary: Let us suppose that F is an injective left D-module.

Then, we have the following commutative exact diagram: V z = 0 = R y ← − y Fq

R.

← − Fp ← − kerF(R.) ← − 0 ↑ V .

• ↑ f ⋆

Fs

U.

← − Fr

S.

← − Fp ← − kerF(S.) ← − 0.

X.

− → 0 = U z ← − z = S y ← − y Moreover, we have: R y = 0 ⇔ U V

• z = 0,

S y = z. General solution: y = u + X z where S u = 0 and U V

• z = 0.

Alban Quadrat Factoring and decomposing linear functional systems

Example: OD system

• Let D = k[t]
• ∂; id, d

dt

• and M = D1×4/(D1×4 R), where:

R =     ∂ −t t ∂ ∂ t ∂ − t ∂ −1 ∂ −t ∂ + t ∂ − 1 ∂ ∂ − t t ∂     ∈ D4×4.

• We obtain the following idempotent:

P =     1 1     ∈ k4×4 : P2 = P.

• We obtain the factorization R = V S, where:

S =     ∂ −t ∂ 1 1     , V =     1 t ∂ 1 t ∂ −1 1 ∂ + t ∂ − 1 1 1 t ∂     .

Alban Quadrat Factoring and decomposing linear functional systems

Example

• Using the fact that we must have Ip − P = X S, we then obtain:

X =     1 1     .

R y = 0 ⇔ y = u + X z : V z = 0, S u = 0.

• The solution of S u = 0 is defined by:

u1 = 1 2 C1 t2 + C2, u2 = C1, u3 = 0, u4 = 0.

• The solution of V z = 0 is defined by: z1 = 0, z2 = 0 and

z3(t) = C3 Ai(t) + C4 Bi(t), z4(t) = C3 ∂ Ai(t) + C4 ∂ Bi(t).

• The general solution of R y = 0 is then given by:

y = u + X z = 1 2 C1 t2 + C2 C1 z3(t) z4(t) T .

Alban Quadrat Factoring and decomposing linear functional systems

Idempotents & Projective modules

• Definition: A left D-module M is projective if there exists a left

D-module N and l ∈ Z+ such that M ⊕ N ∼ = D1×l.

• Lemma: If P ∈ Dp×p is an idempotent, then:

kerD(.P) and imD (.P) are projective left D-modules of rank m and p − m. imD(.P) = kerD(.(Ip − P)).

• Proposition: Let P ∈ Dp×p be an idempotent. 1 ⇔ 2:

1 kerD(.P) and imD (.P) are free modules of rank m and p − m. 2 ∃ U ∈ GLp(D) satisfying U P U−1 =

Ip−m

• ⇒ U = (UT

1

UT

2 )T,

• kerD(.P) = D1×m U1,

imD(.P) = D1×(p−m) U2.

Alban Quadrat Factoring and decomposing linear functional systems

Block diagonal decomposition

• Theorem: Let R ∈ Dq×p, M = D1×p/(D1×q R) and

f ∈ endD(M) defined by P and Q satisfying: P2 = P, Q2 = Q. If the left D-modules kerD(.P), imD(.P), kerD(.Q), imD(.Q) are free of rank m, p − m, l, q − l, then there exist two matrices U = (UT

1

UT

2 )T ∈ GLp(D) and V = (V T 1

V T

2 )T ∈ GLq(D)

such that R = V R U−1 = V1 R W1 V2 R W2

• ∈ Dq×p,

where U−1 = (W1 W2), W1 ∈ Dp×m, W2 ∈ Dp×(p−m) and: U1 ∈ Dm×p, U2 ∈ D(p−m)×p, V1 ∈ Dl×q, V2 ∈ D(q−l)×q.

Alban Quadrat Factoring and decomposing linear functional systems

Example: OD system

• Let us consider the matrix again:

R =     ∂ −t t ∂ ∂ t ∂ − t ∂ −1 ∂ −t ∂ + t ∂ − 1 ∂ ∂ − t t ∂     .

• A projector f ∈ endD(M) is defined by the idempotents

P =     1 1     , Q =     t + 1 1 −1 −t 1 1 −1 t + 1 1 −1 −t t 1 −1 −t + 1     .

i.e., P and Q satisfy: R P = Q R, P2 = P, Q2 = Q.

Alban Quadrat Factoring and decomposing linear functional systems

• Computing bases of the left D-modules

kerD(.P), imD(.P), kerD(.P), imD(.Q), we obtain the unimodular matrices: U =     1 1 1 1     , V =     −1 1 −t −1 1 t t + 1 1 −1 −t −1 1     .

• R is then equivalent to the following block diagonal matrix:

R = V R U−1 =     ∂ −1 t ∂ ∂ −t ∂     .

Alban Quadrat Factoring and decomposing linear functional systems

Example: Saint-Venant equations

• We consider D = Q
• ∂1; id, d

dt

• [∂2; σ, 0] and:

R =

• ∂2

2

1 −2 ∂1 ∂2 1 ∂2

2

−2 ∂1 ∂2

• .
• A projector f ∈ endD(M) is defined by the idempotents

P =      1/2 1/2 1/2 1/2 1      , Q =   1/2 1/2 1/2 1/2   , i.e., P and Q satisfy: R P = Q R, P2 = P, Q2 = Q.

Alban Quadrat Factoring and decomposing linear functional systems

Example: Saint-Venant equations

             U1 = kerD(.P) =

• 1

−1

• ,

U2 = imD(.P) = 1 1 1

• ,

V1 = kerD(.Q) =

• 1

−1

• ,

V2 = imD(.Q) =

• 1

1

• ,

and we obtain the following two unimodular matrices: U =   1 −1 1 1 1   , V = 1 −1 1 1

• .
• We easily check that we have the following block diagonal matrix:

R = V R U−1 = ∂2

2 − 1

1 + ∂2

2

−4 ∂1 ∂2

• .

Alban Quadrat Factoring and decomposing linear functional systems

Corollary

• Corollary: Let R ∈ Dq×p, M = D1×p/(D1×q R) and

f ∈ endD(M) be defined by P and Q and satisfying P2 = P and Q2 = Q. Let us suppose that one of the conditions holds:

1 D = A[∂; σ, δ], where A is a field and σ is injective, 2 D = k[∂1; σ1, δ1] . . . [∂n; σn, δn] is a commutative Ore algebra, 3 D = A[∂1; id, δ1] . . . [∂n; id, δn], where A = k[x1, . . . , xn] or

k(x1, . . . , xn) and k is a field of characteristic 0, and: rankD(kerD(.P)) ≥ 2, rankD(imD (.P)) ≥ 2, rankD(kerD(.Q)) ≥ 2, rankD(imD (.Q)) ≥ 2. Then, there exist U ∈ GLp(D) and V ∈ GLq(D) such that R = V R U−1 is a block diagonal matrix.

Alban Quadrat Factoring and decomposing linear functional systems

Example: Flexible rod

• Let us consider the flexible rod (Mounier 95):

R =

• ∂1

−∂1 ∂2 −1 2 ∂1 ∂2 −∂1 ∂2

2 − ∂1

• .

P =    1 + ∂2

2

−1

2 ∂2 2 (1 + ∂2)

2 ∂2 −∂2

2

1    , Q =

• 1

−1

2 ∂2

• ,

⇒ U =    −2 ∂2 ∂2

2 + 1

−2 ∂2 1    , V =

• −1

2 −∂2

• ,

⇒ R = V R U−1 =

• ∂1

∂1 (∂2

2 − 1)

−2

• .

Alban Quadrat Factoring and decomposing linear functional systems

• V. Implementation: the Maple morphisms package

Alban Quadrat Factoring and decomposing linear functional systems

The morphisms package

• The algorithms have been implemented in a Maple package

called morphisms based on the library OreModules developed by Chyzak, Q. and Robertz:

http://wwwb.math.rwth-aachen.de/OreModules

• List of functions:

Morphisms, MorphismsConst, MorphismsRat, MorphimsRat1. Projectors, ProjectorsConst, ProjectorsRat, Idempotents. KerMorphism, ImMorphism, CokerMorphism, CoimMorphism. TestSurj, TestInj, TestBij.

• QuadraticFirstIntegralConst. . .
• It will be soon available with a library of examples

Alban Quadrat Factoring and decomposing linear functional systems

Conclusion

• Contributions:

We use constructive homological algebra to provide algorithms for studying general LFSs (e.g., factoring or decomposing). We apply the obtained results in control theory.

• Work in progress:

Using morphism computations for factoring and decomposing general linear functional systems, in the proceedings of the Mathematical Theory of Networks and Systems (MTNS), Kyoto (Japan), 2006, rapport INRIA.

• Open questions:

Bounds in the general case. Criteria for choosing the right P. Existence of a solution to the Riccati equation. Formulas for connections. . .

Alban Quadrat Factoring and decomposing linear functional systems